= Lattice model of a modular form
{title2=$V_k$}
Let $\mathcal L$ be the set of <complex lattices>. A weight-$k$ lattice function is a function satisfying
$$
F(\lambda\Lambda)=\lambda^{-k}F(\Lambda)\qquad(\lambda\in\mathbb C^\times).
$$
Set $\Lambda_\tau=\mathbb Z\tau+\mathbb Z$ and $f(\tau)=F(\Lambda_\tau)$. For $\gamma=\begin{pmatrix}a&b\\c&d\end{pmatrix}\in SL_2(\mathbb Z)$, a change of oriented <basis> gives $\Lambda_{\gamma\tau}=(c\tau+d)^{-1}\Lambda_\tau$, hence
$$
f(\gamma\tau)=(c\tau+d)^k f(\tau).
$$
Thus the resulting function has the weight-$k$ transformation law; ordinary invariance under the <modular group> holds only when $k=0$.
Conversely, a function with this transformation law defines $F(\omega_2\Lambda_\tau)=\omega_2^{-k}f(\tau)$ independently of the chosen oriented <basis>. To obtain actual <modular forms>, one must additionally require holomorphy on the <complex upper half-plane> and at the <modular cusp>; homogeneity alone imposes no analytic regularity. For even $k>2$, the <lattice Eisenstein sum> is an example, and its scaling law explains its weight.<Modular Functions and Modular Forms by James S. Milne>[Section 4: modular forms and lattice functions]{ref}
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